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**MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY**

Bank Management, 5th edition. Timothy W. Koch and S. Scott MacDonald Copyright © 2003 by South-Western, a division of Thomson Learning MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY Chapter 9

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Duration GAP …focus on managing NII or the market value of equity, recognizing the timing of cash flows. Interest rate risk is measured by comparing the weighted average duration of assets with the weighted average duration of liabilities. If Asset duration > Liability duration interest rates Market value of equity will fall.

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**Duration versus maturity**

1.) 1000 loan, principal + interest paid in 20 years. 2.) 1000 loan, 900 principal in 1 year, principal in 20 years. int | | | 900+int int |----| | | What is the maturity of each? 20 years What is the "effective" maturity? 2.) = [(900/100) x 1]+[(100/1000) x 20] = 2.9 yrs 1 2 Duration, however, uses a weighted average of the present values.

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**Duration …approximate measure of the price elasticity of demand**

Solve for Price: P -Duration x [y / (1 + y)] x P Price (value) changes Longer maturity/duration larger changes in price for a given change in i-rates. Larger coupon smaller change in price for a given change in i-rates.

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**Measuring duration In general notation, Macaulay’s duration (D):**

Examples face, 10% coupon, 3 year, 12% YTM

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**If YTM = 5% 1000 face, 10% coupon, 3 year, 5% YTM**

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**If YTM = 20% 1000 face, 10% coupon, 3 year, 20% YTM**

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**If YTM = 12% and Coupon = 0 1000 face, 0% coupon, 3 year, 12% YTM**

1000 | | | |

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**Duration and modified duration**

Duration allows market participants to estimate the relative price volatility of different securities: Using modified duration: modified duration= Macaulay’s duration / (1+y) We have an estimate of price volatility: %change in price = modified duration x Dy

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Effective Duration …used to estimate a securities price sensitivity when the security contains embedded options. Effective duration is Where Pi- = price if rates fall, Pi+ = price if rates rise; P0 = initial (current) price; i+ initial market rate plus the increase in rate; i- = initial market rate minus the decrease in rate. Effective duration compares a security’s estimated price in a falling and rising rate environment.

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**Effective duration example: Consider a 3-year, 9**

Effective duration example: Consider a 3-year, 9.4 percent coupon bond selling for $10,000 par to yield 9.4 percent to maturity. The Macaulay’s duration for the option-free version of this bond with semiannual coupons and compounding was calculated to be 5.36 semiannual periods, or 2.68 years at the market rate of 4.7 percent semiannually. The modified duration was 5.12 semiannual periods or 2.56 years. A 30 basis point increase in rate to 5 percent semiannually will lower the price to $9, The callable bond’s effective duration for a 30 basis point (0.30 percent) semiannual movement in rates either up or down is 2.54:

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**Two types of interest rate risk**

Reinvestment rate risk ... the risk that a bank can not reinvest cash flows from assets or refinance rolled over or new liabilities at a certain rate in the future Cost of funds versus the return on assets Funding GAP, impact on NII Price Risk … changes in interest rates will also cause a change in the value (price) of assets and liabilities Longer maturity (duration) larger change in value for a given change in interest rates Duration GAP, impact on market value of equity

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**Reinvestment rate risk and price risk**

If i-rate , then yield from reinvestment of the cash flows and holding period return (HPR) increases. Price risk: If i-rate and you sell the security prior to maturity, the price or value falls, hence HPR falls. Increases in i-rates will improve HPR from a higher reinvestment rate but reduce HPR from capital losses if the security is sold prior to maturity. An immunized security is one in which the gain from the higher reinvestment rate is just offset by the capital loss. This point is where your holding period equals the duration of the security.

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**There are four steps in duration gap analysis.**

Management develops an interest rate forecast. Management estimates the market value of bank assets, liabilities and stockholders’ equity. Management estimates the weighted duration of assets and weighted duration of liabilities. The effects of both on- and off-balance sheet items are incorporated. These estimates are used to calculate duration gap. Management forecasts changes in the market value of stockholders’ equity across different interest rate environments.

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**Duration GAP at the bank**

The bank can protect either the market value of equity (MVE) or the book value of NII, but not both. To protect the MVE the bank would set DGAP to zero: DGAP = DA - u x DL where DA = weighted average duration of assets DL = weighted average duration of liabs

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**Focus on the market value of equity (MVE)**

We know that: DMVE = DMVA – DMVL With DAi = – DAi [ Dy / (1+y) ] Ai and DLj = – DLj [ Dy / (1+y) ] Lj Hence: DMVE = –[DA – (MVL / MVA) DL] [Dy / (1+y)] MVA If we define a bank’s duration gap: (DGAP) = DA – (MVL / MVA) DL, then DMVE = – DGAP [ Dy / (1+y) ] MVA

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**Hypothetical Bank Balance Sheet**

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**Calculating DGAP DA = (700/1000)*2.69 + (200/1000)*4.99 = 2.88**

DL = (620/920)* (300/920)*2.81 = 1.59 DGAP = (920/1000)*1.59 = 1.42 years What does 1.42 mean? The average duration of assets > liabilities, hence asset values change by more than liability values.

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**1 percent increase in all rates.**

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Calculating DGAP DA = (683 / 974) * (191 / 974) * = 2.86 yrs DA = (614 / 906) * (292 / 906) * = 1.58 yrs DGAP = (906 / 974) * = 1.36 years What does 1.36 mean? The average duration of assets > liabilities, hence asset values change by more than liability values.

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**Change in the Market Value of Equity**

Using the relationship: We can define the change in the MVE as: In our case: MVE = (-1.42) x [+0.01 / (1.10)] x 1,000 = -$12.90

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**Positive and negative DGAPs**

Positive DGAP …indicates that assets are more price sensitive than liabilities, on average. Thus, when interest rates rise (fall), assets will fall proportionately more (less) in value than liabilities and the MVE will fall (rise) accordingly. Negative DGAP …indicates that weighted liabilities are more price sensitive than assets. Thus, when interest rates rise (fall), assets will fall proportionately less (more) in value that liabilities and the MVE will rise (fall).

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**An immunized portfolio … What is the minimum risk position?**

To eliminate the risk of changes in the MVE, how much must DA or DL change by? Change DA = -1.42 Change DL = +1.42/u = 1.54

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Immunized portfolio DGAP = 2.88 – 0.92 (3.11) ≈ 0

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**Immunized portfolio: 1% increase in all rates**

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**Stabilizing the book value of net interest income**

This can be done for a 1-year time horizon, with the appropriate duration gap measure: DGAP* MVRSA(1- DRSA) - MVRSL(1- DRSL) where MVRSA = cumulative market value of RSAs, MVRSL = cumulative market value of RSLs, DRSA = composite duration of RSAs for the given time horizon; equal to the sum of the products of each asset’s duration with the relative share of its total asset market value, and DRSL = composite duration of RSLs for the given time horizon; equal to the sum of the products of each liability’s duration with the relative share of its total liability market value. If DGAP* is positive, the bank’s net interest income will decrease when interest rates decrease, and increase when rates increase. If DGAP* is negative, the relationship is reversed. Only when DGAP* equals zero is interest rate risk eliminated. Banks can use duration analysis to stabilize a number of different variables reflecting bank performance.

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**Market value of equity sensitivity analysis**

MVE sensitivity analysis effectively involves the same steps as earnings sensitivity analysis. In MVA analysis, however, the bank focuses on: the relative durations of assets and liabilities, how much the durations change in different interest rate environments, and what happens to the market value of equity across different rate environments.

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**Embedded options sharply influence the estimated volatility in MVE**

Prepayments that exceed (fall short of) that expected will shorten (lengthen) duration. A bond being called will shorten duration. A deposit that is withdrawn early will shorten duration. A deposit that is not withdrawn as expected will lengthen duration.

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ABC's market value of stockholders' equity Market value/duration report as of 12/31/01 most likely rate scenario-base strategy (assets) Assets

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ABC's market value of stockholders' equity Market value/duration report as of 12/31/01 most likely rate scenario-base strategy (liabilities) Liabilities

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**Duration Gap for ABC’s MVE**

Mkt value of assets = 1,001,963 Duration of assets = 2.6 Mkt value of liabilities = 919,400 Duration of liabilities = 2.0 Dur Gap= 2.6 – (919,400 / 1,001,963) * 2.0 = yrs Example: A 1% increase in rates would reduce MVE by 7.2 million = (0.01 / ) 1,001,963 Recall that the average rate on assets is 6.93%

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**Sensitivity of economic value of equity (MVE) versus most likely (zero shock) interest rate scenario**

Sensitivity of Economic Value of Equity measures the change in the economic value of the corporation’s equity under various changes in interest rates. Rate changes are instantaneous changes from current rates. The change in economic value of equity is derived from the difference between changes in the market value of assets and changes in the market value of liabilities.

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**Effective “duration” of equity**

By definition, duration measures the percentage change in market value for a given change in interest rates Hence, a bank’s duration of equity measure the percentage change in MVE that will occur with a 1 percent change in rates: Effective duration of equity yrs. = 8,200 / 82,563

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**Asset / liability sensitivity and DGAP**

Funding GAP and Duration GAP are not directly comparable. Funding GAP examines various “time buckets” while DGAP represents the entire balance sheet. Generally, if a bank is liability (asset) sensitive in the sense that net interest income falls (rises) when rates rise and vice versa, it will likely have a positive (negative) DGAP suggesting that assets are more price sensitive than liabilities, on average.

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**Advantages of DGAP over Funding GAP**

DGAP analysis has the advantage of focusing on all cash flows from the underlying assets and liabilities and not just cash flows that are expected to arise over short time intervals. Interest rate risk can be summarized in one measure for the entire portfolio.

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Speculating on DGAP It is difficult to consistently alter either GAP or DGAP on the balance sheet and increase earnings or the market value of stockholders' equity. Whenever management chooses to change asset and liability maturities and/or durations in anticipation of rate changes, it is placing a bet against forward rates from the yield curve.

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**Gap and DGAP management strategies: what are your bets?**

Cash flows from investing $1,000 either in a 2-year security yielding 6 percent or two consecutive 1-year securities, with the current 1-year yield equal to 5.5 percent. | | | 2-year security: ----$60----|----$ $120 at 6% per yr 1-year security: ----$55----|----- ? $120 another 1-yr sec. It is not known today what a 1-year security will yield in one year. For the two consecutive 1-year securities to generate the same $120 in interest, ignoring compounding, the 1-year security must yield 6.5% one year from the present. This break-even rate is a 1-year forward rate, one year from the present: 6% + 6% = 5.5% + ? with ? = 6.5 percent

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Interest Rates and the Business Cycle The general level of interest rates and the shape of the yield curve appear to follow the U.S. business cycle. In expansionary stages rates rise until they reach a peak as the Federal Reserve tightens credit availability. P e a k S h o r t - T e r m R a t e s ) t L o n g - T e r m R a t e s n e c r e P ( E x p a n s i o n s e t a C o n t r a c t i o n R E x p a n s i o n t s In contractionary stages rates fall until they reach a trough when the U.S. economy falls into recession. e r e t T r o u g h n I T i m e

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**MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY**

Bank Management, 5th edition. Timothy W. Koch and S. Scott MacDonald Copyright © 2003 by South-Western, a division of Thomson Learning MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY Chapter 9

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